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Extra info for Dynamic Programming and Partial Differential Equations
Much of what follows in the succeeding chapters will be based on matrix versions of these results. EXERCISES 1. Prove (9). 2. Find the recurrence equations for the coefficients r k , s, and t k . Use these results to obtain a recurrence equation for the x k . BlBLlOG RAPHY AND COMMENT See R. Bellman, Introduction to the Mathematical Theory of Control Processes I : Linear Equations and Quadratic Criteria, Academic Press, New York, 1967. Section 7 . See R. Bellman, “Functional Equations in the Theory of Dynamic Programming XVII : Minimum Convolutions and Green’s Functions,” J.
2). The solution of linear differential equations of this nature can be effected by means of the solution of M-dimensional linear algebraic systems. The associated variational problem is the minimization of The positive-definite nature of this functional ensures the existence and uniqueness of the solution of (1) and (2). 14. Irregular Grid There is no necessity, of course, to employ a regular grid. ,Yn-l,Yn (1) 9 where 0 = y o < y1 < y 2 < ... ,yn. (2) One way to determine the quantities ivi and yi is to require that the formula be exact for polynomials of degree n or less.
This transformation, of course, is the beginning of the real problem, that o f obtaining useful analytic and computational results from the linear system. 8. Rectangular Region Let us see what this involves for a rectangular region (Fig. 3). Let integers M and N be chosen and the positive quantities A and 6 be determined by M A = a, N 6 = 6. , N . q(rnd, h ) . 'obr7pl We suppose t h a t g ( s , y) is continuous so that g(0, 0) and g ( m A , N 6 ) are unambiguously defined by the foregoing. 01 29 10.
Dynamic Programming and Partial Differential Equations by Angel